Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line

Abstract

For the following class of partial neutral functional differential equations \[ \begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), \\ u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases} \] we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.